Market Equilibrium with Private Knowledge: An Insurance Example

Kunreuther, H.C. and Pauly, M.V.

IIASA Working Paper

WP-82-058

June 1982

Kunreuther, H.C. and Pauly, M.V. (1982) Market Equilibrium with Private Knowledge: An Insurance Example. IIASA

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MAFUQT EQUILIBRIUM WITH PRIVATE KNOWLEDGE: AN INSURANCE EXAMPLE

Howard Kunreuther Mark Pauly

J u n e 1 9 8 2 WP82-58

Working Papers are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute or of its National Member Organizations.

IXTERNATIONAL IKSTITUTE FOR AF'PLIED =STEMS .ANALYSIS 2361 Laxenburg. Austria

CONTENTS

I. INTRODUCTION

11. MARKET EQUILIBRIUM WITH ONE FULLY INFORMED FIRM

I n s u r e r ' s P o t e n t i a l S t r a t e g i e s S t a b i l i t y of E q u i l i b r i u m W e l f a r e E f f e c t s

111. INFORMED FIRMS: LEARNING OVER TIME

N a t u r e of E q u i l i b r i u m An I l l u s t r a t i v e E x a m p l e O b t a i n i n g V e r i f i e d I n f o r m a t i o n

I V . A MYOPIC MULTI-PERIOD MODEL

V. CONCLUSIONS AND EXTENS IONS

REFERENCES

APPENDIX A

MARKET EQUILIBRIUM WITH PRIYATE KNOWLEDGE: AN INSURANCE EXAMPLE^

Howard Kunreuther and Mark Pauly

I. INTRODUCTION

The effect of asymmetric information between buyers and sellers on

product quality was first explored by Akerlof (1970) in his pathbreaking

paper on the market for "lemons." He showed that if all purchasers have

imperfect information on quality, then a market for the product may not

exist, or if it does function it may not be efficient. These results have led

to a number of papers concerning insurance and labor markets under dif-

ferent assumptions regarding how agents discriminate between "pro-

'The research reported in this paper is partially supported by the Bundesrninisterium f i k Forgchung und Technologic, F.R.G., contract no. 321/7501/RGB 8001. While support for this wark is grateiully acknowledged, the views expressed are the authors' own m d are not neces aarily shared by the sponsor. We are gratetul to Zenan Fortuna and Serge Medow for compu- tational assistance and to David Cummins and Peyton Young and the participants in the Conference on Regulation of the International tnstitute of Management, Berlin, July, 1081, especially Jdrg Rnsinger and Paul Kleindorfer. for helpful comments and suggestions. An earlier version of this paper is included in the conference proceedings.

ducts" of varying quality. (See Pauly 1974, Rothschild and Stiglitz 1976,

Wilson 1977, Miyasaki 1977, and Spence 1978).

These treatments assume that all of the imperfectly informed agents

have identical levels of knowledge of product quality. In contrast, this

paper will consider situations where some agents learn over time about

the quality of a particular good. However, this knowledge is private or

agent-specific and may be costly for others to obtain. For example, firms

may learn about the differential skills of their labor force by observing

their productivity; other firms do not have easy access to this informa-

tion. Insurance firms learn about the risk characteristics of their custo-

mers by observing claims records; they will not share these data cost-

lessly with their competition. We are interested in characterizing the

nature of the market equilibrium when agents have such private

knowledge on the endowed qualities of a good. Our analysis is undertaken

in the context of insurance markets, although it is applicable to those

situations where sellers cannot easily communicate their product's spe-

cial qualities to prospective buyers even though the current purchasers

have at least partially observed these features.

The following problem is first analyzed in detail. Suppose that a set

of customers has been with a specific insurance firm for t years, durmg

which time the firm has collected information on their claims experience.

The insurer naturally will not make these data available to other firms,

and consumers are unable to furnish verified hstories. Not having direct

knowledge of each customer's risk class, the insurance firm utilizes

claims data to set premiums. What is the schedule of profit-maximizing

rates at which no customer will have an incentive to purchase insurance

elsewhere in period t + l?

We consider two polar cases with regard to the assumption made

about firm behavior. In one case, we assume that the firm has n o

j w e s i g h t , so that it sets prices to make non-negative expected profits in

every period. In the other, we assume that the firm has pe r fec t fo res igh t ,

in the sense that it maximizes the present discounted value of the

expected profit stream over the planning horizon. We assume in each

case that consumers choose the firm making the most attractive offer in

the current period. In this paper, consumers do not have the foresight to

consider the stream of premiums that will be charged in the current

period and all future periods. This assumption therefore represents one

polar case, with perfect consumer foresight models such as those of

Dionne (1981), Radner (1981), and Rubinstein and Yaari (1980) a t the

other extreme.

We refer to the situation in which neither firm nor customers have

foresight as s ing le - per iod equi l ibr ia , since firms can change their price

from one period to the next and consumers are free to stay or leave as

they see fit. We refer to the situation in whch firms maximize &scounted

expected profits but consumers choose only on the basis of current

period premiums as m y o p i c mu l t i - per iod equ i l ib r ium. 2

The paper is organized as follows. We first begin at the end, so to

speak, by considering in Section I1 a static model in whch the firm

currently selling insurance to individuals has perfect private knowledge

'one of the purposes of experience rating is to cope with problems of moral hazard. This pa- per does not answer the question as to whether a premium adjustment process ce.n eliminate or substantially reduce moral hazard. The paper also does not consider the possibility of re- quiring individuals to state their probability of loss and using experience to "punish" those who misstate (cf. Radner 1881, Rubinstein and Yaari 1880).

about each person's risk class. We show how the premiums charged are

nevertheless constrained by other insurance firms. Section 111 develops a

model in which firms use i .o rmat ion from the claims experience of the

insured in a Bayesian fashion to adjust individual premiums to experi-

ence. We show that in the single-period equilibrium model, the resulting

premium schedule yields positive expected profits and monopoly distor-

tions even i f entry by new firms into the market is completely free. Profit

or rate regulation would be a natural remedy if reality approximated this

equilibrium. We further consider briefly the impact on the single period

equilibrium of permitting customers to buy verified information on their

experience. This would include purchase of data on premium classifica-

tions or claim records.

Section lV shows that in a myopic multi-period equilibrium, expected

profits are zero with free entry, but price distortions remain. Premiums

are generally below expected costs in the early periods, but eventually

rise to exceed expected costs. The concluding section suggests applica-

tions and extensions of the analysis.

IL MARKET EQUILIBRIUM WITH ONE FULLY INFORMED F'IRM

Our world consists of two types of consumers. Every consumer faces

the possibility of an identical single loss (x) whch is correctly estimated

and which is independently distributed across individuals. Each consu-

mer of type i has a probability of a loss Q i , i = H , L for the high and low

risk group respectively ( a H > a L ) . The consumers correctly perceive

these values of Qi. The proportion of h g h and low risk consumers in the

population is given by NH and NL respectively. Type i's preferences are

represented by a von Neumann-Morgenstern utility function, Ui, and each

consumer determines the optimal amount of insurance to purchase by

maximizing expected utility E ( U i ) . The insurance industry consists of n

firms, all of whom correctly estimate X and the average probability of

loss correctly.

We initially assume that each consumer has been insured by the

same firm for a sufficiently long period of time that the insurer has col-

lected enough information through claim payments and other data to

specify ai exactly. The remaining n-1 firms in the industry cannot

determine whether individuals insured by others are h g h or low risk peo-

ple; an insured's past history does not become common knowledge.

If firms have information on the risk class of their clients they can

charge differential premiums to high and low risk individuals; other firms

in the industry are forced to charge the same premium to both groups

because they cannot distinguish high risks from low risks. However, each

firm does know how many periods the individual has been in the market.

including whether he is a new customer.

Insurers' Potential Strategies

We now characterize the strategies available to insurers and consider

the possibility of equilibrium. With regard to a particular client, it is use-

ful to think of firms as either being "informed," i.e, having sold a policy to

an individual in the previous period, or "uninformed," i.e., treating the

client as a customer new to that firm. Each informed insurer offers a

per-unit premium, Pi , to all individuals in risk group i, without specifying

the amount of coverage, Qi, which an individual may purchase, except

that 0 < Qi 5 X . Insurers who are not informed about a set of individuals

charge the same price to all of them. Consider first the situation of a

representative uninformed firm. It knows that each consumer has the

insurance demand curve:

which is derived from constrained utility maximization. Since the unin-

formed firm cannot distinguish among risks, it will have to set

PH = PL = P . In a free-entry world with firms that maximize expected

profit E(rr), the breakeven premium P* for such uninformed firms would

be given by the lowest value of P such that:

where QL is the total amount purchased by each L and QH is the total

amount purchased by each H at the uniform premium P*. If equilibrium

exists, the low risk group will subsidize the high risk group and purchase

partial coverage QL < X, while Q h risk individuals will purchase full cov-

erage, QH = X, at subsi&zed rates. 3

The informed insurer can use his exact knowledge of each present

customer's G i , i = H , L , to set rates tailored to each customer's experi-

ence. For u h risk individuals, the informed firm w d never charge less

'When the only value of P which satisfies (2) is P* = a H , then QL = 0, and the market will only provide coverage to high risk individuals. This is a case of market failure, since low risk individuals cannot purchase insurance due to imperfect information by firms.

than i P H . For low risk individuals, the rate it will charge will depend on

the premium charged by uninformed firms. The informed firm maxim-

izes expected profits by charging low risk individuals a price Pi that is a

little less than the uniform price offered by the uninformed firm to all

purchasers of insurance. The informed firm then attracts all low risks,

sells each of them QL units and makes profits of (PL - i P L ) ~i per type L

person.

If, for example, uninformed firms are charging P* + 6, the inlormed

firm will want to charge its low risk customers the lower of one of two

rates. It will either charge P*, or it will charge pL, the premium which

would maximize profits on low-risk insureds if the firm were a monopolist.

At the other extreme, if uninformed firms are charging BH to everyone,

then the informed firm will charge either aH - E or pL, whichever is less.

But just as the informed firm's optimal pricing strategy depends on

the strategies selected by uninformed firms, so does an uninlormed

firm's strategy depend on what the informed firm is doing. The strategic

combinations and payoffs are shown as the payoff matrix, in Figure 1,

with the upper expression in each labeled cell 1...4 being the payoff (pro-

f ib) to the informed firm (I), and the lower expression the payoffs to the

uninformed firm (U). When one type of firm obtains no business, and all

customers purchase from the other type of firm, a profit level of zero is

entered. Here we are assuming that both P* and aH are less than pL. 4

I %f pL is less than P*, then the informed firm will always charge PH = a H and = pL making positive profits. Uninformed firms will not obtain any business no matter what they do.

-

Figure 1. Payoff Matrix for Informed and Uninformed Firms.

Uninformed Firm (U)

Informed

Stability of Equilibrium

We will now show that there is no stable Nash equilibrium where there

are both fully informed and uninformed firms. The argument is simple. If

U (uninformed) firms chose PU = aH, then I (informed) should choose

pj$ = aH - e to maximize profits. ~ u t if I chooses aH - e, there exists

some smaller P' + 6 at which U can make positive profits, while 1 gets no

business and makes zero profit. But if U charges some P' + 6, I should

charge a little less (e.g., P*). Then I makes positive profits, but U suffers

a loss. To prevent t h s loss, U must charge at least QH. But then I should

charge QH - E , etc. If there are many players, the absence of a Nash

equilibrium makes stability unlikely.5

pi = P' NLQL (p' - @L) > 0

p V = p ' + 6 < a H

5 ~ o t e that, from the viewpoint of a single unintormed firm the maximum value that 6 can take in cell 3 in Figure 1 depends on what the firm assumes that the other uninformed firms wil l do. Li they continue playing strategy, then the shgle uninformed firm can charge anything less than @H - & and capture all the business with a large profit. Lf each unin- formed firm assumes the other uninformed firms will match its prices, then profits will be lower.

pV = aH

What other concepts of equilibrium might apply here? If both parties

followed maximin strategies, the outcome would be in cell 2, with the

strategy ( P L = P* , Pfi = @ H j for the informed firm, and [ P V = B H { for

the uninformed firm. In this cell, the uninformed firm is sure that it will

not lose money (although it will not make profits either). The informed

firm guarantees itself positive profits. Thus, in a single-play context, or

with a small number of players, we might expect the outcome to be in cell

2.

Another possibility, already used in the literature on insurance

markets and imperfect labor markets, is the concept of Wilson Equili-

6 brium. A given set of actions is a Wilson equilibrium if no firm can alter

its behavior (i.e., propose a different premium) that will (a) earn larger

positive profits immediately, and (b) continue to be more profitable after

other firms have dropped all policies rendered unprofitable by the initial

firm's new behavior. Is the pair ( P L = P* , Ph = ' P H j and [ P V = i P H j a Wil-

son equilibrium? The alternative strategy for the informed firm is to set

{PL = i P H - E , PA = aHj. Tbs earns it larger profits and does not cause

the uninformed firms to lose money if they maintain their same policy as

before. However, an informed firm's charging [ PL = GH - E Pfr =

would permit uninformed firms to make positive protits by switcbng to

Pu = P* + 6 ; this change reduces the informed firm's profit to zero.

Thus, li we substitute the notion "rendered less profitable" for "rendered

unprofitable" in part (b) of the above definition, then cell 2 does qualify as

a Wilson equilibrium.

'1t was proposed by Wilson (1977) and has been utilized by, among others, Miyadci (1977), and Spence (1 W8), to characterize equilibrium.

An alternative equilibrium concept which leads to the same conclu-

sion is based on a Stackelberg leader-follower model. It seems reasonable

to suppose that the (single) informed firm will play the leadership role.

We will assume that the informed firm always sets PA = aH. The reaction

function for the uninformed firm is P V = j' (PL), and the informed firm

therefore maximizes its expected profit (TIf):

If the informed firm sets Pi = P', then PV = j' (P i ) = aH, and the

informed firm makes positive profits of QL (P' - aL) on each type-L per-

son. If the informed firm sets PL = aH - E , then PV = j' (PL) = P' + 6,

and l l f is zero. Hence, maximization of (3) requires Pi to be P', and the

Stackelberg equilibrium is given by cell 2.

To summarize, there are two conclusions based on the above discus-

sion: 7

(1) No single-period equilibrium exists, or

(2) A single-period equilibrium is represented by

= P', PL = for informed firms. 1Pv = aHJ for unin-

formed firms, with all business going to informed firms.

In what follows, we adopt the second conclusion by assuming that the

mformed and uninformed firms behave in a Stackelberg fashion, with the

informed firm as the leader and the uninformed firms as the followers.

This equilibrium is also achieved i f one assumes that either firm follows a

policy that maximizes the minimum profit they could attain no matter

7 ~ e have not considered the possibility of mixed strategies.

what uninformed firms did, or that the modified definition of a Wilson

equilibrium is appropriate.

Welfare Effects

In the no-information case, the equilibrium premium is PC for both

high and low risks. Compared to the no-information case, perfect private

knowledge for just one firm leads generally to no gain in welfare for any

insured person. All of the gains from information go to Wormed

insurance firms as positive long-run profits. In the special case where low

risk individuals are charged the monopoly price (i.e., Pi = pL), the low-

risk class benefits by the amount that the premium is below P*. Even

then, the higher risk consumers are made unequivocally worse off with

perfect private knowledge, since the price they pay increases from P* to

$*. Moreover, the positive profits being earned by informed firms are

not eroded by entry, since new firms are by definition uninformed ones.

Nature of EQuilibrium

We now turn to the more general case where firms learn over time

about the characteristics of their customers through loss data. Initially

each firm only knows from statistical records that the proportion of b g h

and low risk individuals in the insured population is given by NH and NL

respectively. It does not know whether an individual is in the H or L class

but does know how many periods each potential customer has been in the

market (e.g., all 20 year old males are assumed to have been driving

8 legally since age 16) . Any new customer would be offered a premium, P * ,

which is defined as before so that

E ( T ) = N ~ ( P * - @ ~ ) QL + NH ( p * - a H ) QH = 0 (4)

That is, the insurer prices so as to yield expected profits of zero on all

new business. 9

During each time period, we assume that an individual can suifer at

most one loss, which will cause X dollars damage. Any time a claims pay-

ment is made, this information is recorded on the insurer's record and a

new premium, which reflects his overall loss experience, is set for the

next period. As before, we are assuming that informed firms do not dis-

close their records to other firms. Individuals who are dissatisfied with

their new premium can seek insurance elsewhere. Other firms will not

have access to the insured's record and hence cannot verify whether an

applicant has had few or many losses under previous insurance contracts.

The informed firm uses a Bayesian updating process in readjusting

its premium structure on the basis of its loss experience. Consider all

customers who have been with the same insurance company for exactly t

periods. They can have anywhere from 0 to t losses during this interval.

The premium charged for period t + 1 to individuals with j losses during

a t period interval is P;,j = 0 . . . t . lo Firms with loss experience data

'In this sense, a firm can distinguish between new arrivals to the market and customers ormerly insured by other firms.

BTh s seems to be the rule that actuaries are instructed to follow in an experience rating context. For example, the premium in any one year is supposed to be the previous year's premium plus a "bonus" G, where G is defined as:

and P is the expected value of losses, c i s the actual amount of loss in the previous period, 1 is the "safety loading (including normal profit) and k is a fraction less than one. In the ini- tial period when G = 0, actuaries will recommend that the premium equal ( 1 + l ) P (See

ard, Pentikainen, and Pesonen 1979). eh e are assuming that losses for an individual are independent of previous experience so the premium at the end of t is determined only by the number of claims.

will set each premium P$ so that they maximize expected profits, subject

H to the constraint that customers remain with them. Let wko and woo be

the respective probabilities that an individual is in the low and hlgh risk

class when the firm initially insures him. We can update these probabili-

ties by using Bayes' procedure. If a customer has suffered exactly j

losses in a t period interval then we define wjt, i = L ,H as the probability

L that he is in the ith risk class, where w# + wjt = 1.'' The premium set

for each loss classification will also be determined in part by the relative

values of wjt ,i = L,H. As j increases so does the probability that the

individual is in the high risk class. Hence, w$ > GI,t, j = 1 . . . , . t .

Suppose, for example, an informed firm offers a set of ?remiurns

tpitj, with pJ; increasing as j increases.12 An uninformed firm which

charged a lower premium than pit in any period would attract all custo-

mers with j or more 10sses. '~ The proportion of high and low customers

in its portfolio would be given by

where sk = probability of a person suffering exactly k losses in a t period

''we determine wjt as foUows. Let = probability that an individual experiences j losses in t periods, if he is in risk class i. Specifically,

Using Bayes formula

'%e nll show below that Pjt increaser as j increases. 13we are assuming no transaction costs for insured individuals to switch firms.

interval. In other words, wlt is a weighted average over the loss range

j . . , t . Since w$ increases with j we know that w$>w$ for all

j = 1 . . . t - 1 and W [ = 4. The minimum premium (P];) at which expected profit equals zero for

uninformed firms is given by:

H wjt ( p j ; - @ H ) Q; + W$ (pj ; - @ L ) Q,L~ = o (5)

where Qjt is demand for group i given a premium We know that P];

increases with j since W$ increases with j. Hence any new firm which

sets P = P; attracts only customers with j or more losses and makes

zero expected profits. So (5) correctly describes the minimum level of

premiums that uninformed firms can charge.

If the informed firm sets Pjt = P; -E for d y those customers who

have suffered exactly j losses, then these individuals will still prefer the

informed firm. Its expected profits are given by

For sufficiently small E , expected profits in (6) are positive for

j = 1 . - t - 1 since w$ is less than w;. For j = t , as r -. 0, expected

profits by definition will also approach zero since w; = W&

To determine the premium structure, an informed firm will also have

to find the monopoly premiums i p j t 1 for each j = 0 . . t , which maxim-

ize E ( n j t ) . I t will maximize expected profits for each loss category if it

then sets premiums ( p i ) as follows

pj; = min (P]:; - E . B ~ ~ ] . j = 0 . * . t

The structure of the premiums is thus similar to that in the case of per-

fect private knowledge outlined above; profits will be lower because firms

must now use claims information to categorize their customers and

hence will misclassify some of them. Aggregate expected profits for each

period t are given by

An Illustrative Example

A two-period example using a specific utility function will help to

fllustrate the meaning of learning from loss experience. The appendix

describes the basic form of this problem for the exponential utility func-

tion U ( Y ) = - ~ - ~ . Consider the specific case where

@H = ,3, @L = , I , X = 40, c = .04, and NL = NH = .5. Then the equilibrium

premium in the first period, obtained by solving equation ( 4 ) , would be

P* = .254. Table 1 illustrates how one calculates the weights for deter-

mining the optimal premium structure at the end of period 1 when

j = 0 QT 1, and Figure 2 details the optimal rate structure at the end of

period 1 .

The optimal premiums are pil = ,254 and P ; ~ = .288 since

pol = p l l = ,495. The premium charged to the group suffering one loss

(P ; , ) , yields E ( l l l l ) = 0 since P ; ~ = P ; ' ~ , and urK = W E . Expected pro-

fits for the "zero loss" class is given by (6) and is

E(no1) = .5625 ( .254- . lo) 12 + .4375 ( .254-.30) 40 = .23.

Aggregate expected profits for period 1 are given by (7 ) and in t h s case

are E (ml) = .8( .23) = . l a .

- 16-

Table 1: Calculation of Weights wjl and w:, i = L .H for Two Period Model.

7.69 12

Figure 2. A Two Period Example Based on Loss Experience.

What effects do experience rating have on consumer well-being in

this example? In the absence of any information, both high and low risks

would have been charged .254 in each of the two periods. When informa-

tion is obtained through experience, those individuals with no losses are

charged the same rate as initially, .254, but the others with one loss are

charged ,288. Thus both high and low risk customers are made either no

better off or worse off if firms can generate information. In contrast, it

the firm would have charged breakeven prices, its premium would have

been .237 to individuals with zero losses.

As a customer's life with the company increases, then he faces a

larger number of rate classes reflecting possible outcomes. Firms make

the largest profit on those insured individuals who experience the fewest

losses. In the limit as t + m, all customers will be accurately classified

and we have the case of perfect private knowledge. Figure 3 graphically

depicts how aggregate expected profit changes over time as a function of

the proportion of low risk customers in the population. As NL decreases

then the informed firm's profit potential decreases since a larger propor-

tion of individuals will suffer losses.

Obtaining Verified Information

The problem in achieving optimality arises, of course, because

informed firms--the ones from whch the consumer is currently

purchasing--price so as to obtain positive long-run expected profits. A

natural response of low risk consumers facing such a situation is to seek

some way of provldlng reliable information on their status to other

insurance firms. There are two ways in whch such data might be

Aggregate Expected Profit ( E (ITt) )

0 6 12 18 24 30 Time

Figure 3. Aggregate Expected Profits [E(nt)] as a Function of Proportion of Low Risk Customers (NL) and Time ( t ) .

disseminated: (1) Consumers might provide verified information on their

actual number of losses (claims), and/or (2) Consumers might provide

verified information on the size of their premium bill for a given level of

insurance, since this is a perfect indicator of the risk class into which

they are being placed by their current insurer.

We would expect that consumers will find it difficult and costly to

undertake either of these actions. For one thmg, the current insurer has

an incentive to conceal its claims and premium data. For another thing,

purchasers of insurance who have had unfavorable loss experiences may

try to represent themselves as belng a better risk by using techniques

such as bogus invoices, or applying for insurance right after an accident

but before a new bill is issued. Note that the informed firm will not

discourage these actions, because such behavior makes it more difficult

for customers with good experience to communicate their status reliably.

The cost of providing reliable information by insured individuals will

still permit the original insurer to earn some positive profits and the

above models would still be relevant in determining what rate structure

could be set by informed firms. One could formally incorporate the costs

of communicating verified information into a more general model of the

choice processes 01 insurers and insured. Profits would then be limited

by the alternatives available to consumers for purchasing verified infor-

mation.

AT. A MYOPIC MULTI-PERIOD MODEL

We now investigate the consequences of changing the assumption

that there is no insurer foresight. We consider a model in which firms

look beyond current period losses to potential future profits. Firms are

therefore assumed to be concerned with the present discounted value of

the profit stream they expect to earn. But purchasers are still assumed

to be myopic, in the sense that they choose which insurer to patronize by

looking only a t current period prices and selecting the firm with the

lowest current premium. If the firm is willing to tolerate negative

expected profits for a while in order to attract customers and observe

their claims experiences, it can then use this information to make posi-

tive expected profits in the future to offset (in present value terms) the

initial losses.

It is easy to see that the "single-period" premium schedule IPS] may

then not be an equilibrium. On the one hand, a firm that charged less

than P: in the initial period would have an expected loss in that period;

on the other hand, it would have the opportunity to observe which indivi-

duals did and did not have losses during that period. If it used that infor-

mation to charge the schedule P; in subsequent periods, the present

discounted value of the profit stream associated with this pricing policy

could be sufficient to offset the initial expected losses. Hence, a new

schedule, with the lower Po,, would dominate the single-period equili-

brium schedule.

What new set of premiums would represent a n equilibrium schedule?

I t would be one where, for all t and j , there would be no opportunity for a

previously uninformed firm to enter and earn positive expected profits.

To simplify the explanation of how this schedule is derived, we assume an

interest rate of zero, so as not to be concerned with discounting. We

assume that the firm which has attracted a customer in period 0 will want

to set its premiums for all future periods up to the end of the planning

horizon T so that no firm entering the market in later periods can attract

any set of its customers and make a stream of profits whose sum is posi-

tive. That is, it will want to set P T s o that

lor all j . Here ~ T i s also the price that the new entrant would charge. 14

14we will assume that purchasers buy all of their insurance from a single firm. Alternatively, r e could have assumed that each firm receives a constant share of an insured's business in every period, and that all firms are aware of this fact.

The procedure in constructing a set of premiums IP;? requires one

to start at period T and work backwards. Any dninformed firm who

enters the market at the beginning of period T must break even, because

there is by definition no future period in which losses can be recouped.

Hence. P; = PiT for all j. Now consider period T - 1. If a firm entered

in this period it could observe the experience of its customers for one

period and make profits on on all those individuals who did not have a loss

during this period.

The expected profits in period T are derived using the same type of

Bayesian updating procedure described in Section 111. In order to prevent

new entrants from coming into the market in period T - 1, the informed

firm must set its premium in period T - 1 sufficiently low so that a poten-

tial new entrant would suffer a loss just a little larger than the profit he

would earn in period T . As in the single period equilibrium model, there

will be a different premium for each value of j. This set of policies

l~c~~~j would then be the equilibrium schedule for the fully informed

firm.

The same type of reasoning is utilized to compute the equilibrium set

of premiums for period T-2. In this case a potential entrant who attracts

customers can make profits in periods T-1 and T by utilizing claims

information on their insured population. The informed firm will then have

to set l~T~-~j at levels which erase all these potential profits of a new

firm. The same process is repeated sequentially for all periods through

t = O .

To illustrate differences between resulting premiums in the single

period equilibrium and myopic multi-period equilibrium cases we con-

sider an example with T = 5. Table 2 compares the set of premiums and

expected profits for the two models. In the single period equilibrium

model the informed firm's premium (P,L,) star ts off equal to the average

actuarial value (P ' = ,254) and increases above this level for customers

who experience losses. In the myopic case, the initial premium, P:, is

less than P ', and increases over time whether or not the person suffers a

loss.15 As t approaches T, the premiums for the two types of equilibria

converge as expected. In the single-period case, the stream of profits is

positive, in all periods; in the multi-period myopic case the firm suffers

losses in the early periods recouping them in later periods so that the

expected stream of profits is zero.

Table 2 reveals that there is a perversity and allocative inefficiency

in the multi-period myopic case. Consumers are undercharged in the

early periods but will find that their premiums are raised even if they are

accident free. Persons nearing the end of their risk horizons (e.g., the

aged who will only be driving for a few more years) will tend to be over-

charged for insurance, whereas the young will tend to be undercharged.

Hence, consumers will tend to over-purchase insurance in the early

periods, and under-purchase insurance in the later periods. If regulation

could be used to bring premiums closer to the actuarial values, there

would be a welfare gain.

'Tt is theoretically possible for consumers inhally to be charged a negative premium to at- tract them to the insurance company so that they could be charged higher premiums as t increases. Tn this case, individuals could be given a free gift for taking out insurance, in an analogous fashion to the approach used by savings banks to attract new accounts.

Table 2. Comparison Between Premiums and Expected Profits for Single Period Equilibrium and Myopic Multi-Period Equilibrium Schedule

for Five Period Problem.

Single Period Equilibrium

Myopic Multi Period Equilibrium

Period I

Y Number

4

.254

.39

.283

.08

.295

.01

.299

.oo

.30

.OO

.48

losses j Of \I

5

.254

.39

.281

.12

.294

.O 1

.298

.oo

.30

.OO

.30

o

.52

2

.254

.30

.286

.02

.296

o

.32

1

.254

.18

.288

0

.18

3

.254

.36

.285

.04

.296

.OO

.299

0

.40

0

.254

0

0

'Period

Y Number

0

1

2

3

4

5

~(n:)

"o't

E (no't )

Prt

E(n;J

p i t

E(nit)

p i t

p4't

E(nft

" i t

E (nzt )

V. CONCLUSIONS AND EXTENSIONS

Our results have some important implications for the notion that

development of "reputations" for quality can over time alleviate the prob-

lem of agent ignorance (Akerlof 1970). In both of our models, there is an

incentive to keep information about quality private, even if the explicit

cost of communicating it to others is low. In our first model, the agent

with private knowledge loses monoply rents by communicating this infor-

mation to others. In the second model he earns no rents in the long run;

however, he would impose losses on himself if he initially followed the

loss-leader strategy but then communicated the information he had

gained from claims experience before he had time to recoup his losses.

Conversely, even if he "promised in the initial period to communicate

the truth, he would always gain from concealing or corrupting informa-

tion which identifies the good (high quality) risks. These disincentives to

communicate reliable information are greater if the item being tran-

sacted is bought in lumpy amounts, as in the case of a worker's services;

the employer who learns which of his employees are of higher produc-

tivity will be downright reluctant to communicate that knowledge to other

potential employers.

m l e "friends and neighbors" do sometimes communicate informa-

tion about product quality (good restaurants, good doctors) and whle

employers do write letters of reference for good employees, our models

suggest that such behavior is not likely to occur in all circumstances.

Even where concealing private information ends up benefiting none and

harming all, it will still be difficult tor the market to break away from

such an equilibrium.

The most obvious extensions of these models is to permit consumers

to be less myopic. If consumers do have foresight, then they may want

the insurer to agree in the initial (purchase) time period to provide accu-

rate information on future loss experience. The mere availability of such

information would, in itself, be sufficient to eliminate either monopoly

profits or the zero-profit distortions in the multi-period myopic model.

Guaranteeing that such information is provided is not easy, of

course, since the low risks must not only ensure that accurate informa-

tion on his own experience is provided but also that accurate (unfavor-

able) information is provided on the experience of those h g h risks who

are thinking of switching to another firm. That is, he must monitor the

accuracy of all information provided. For example, in a labor market

application of our theory, a high productivity worker must not only verify

that his employer will provide him with a good and true recommendation;

he must also verify that poorer quality workers are being furnished bad

recommendations or references.

If the worker has foresight, it is easy to see that he will be con-

cerned, in the initial period, with the schedule by which his future premi-

ums will be adjusted as a result of his future experience. Different risk

types mght be expected to select different schedules and Dionne's work

(1981) shows that it is possible in a monopoly context to find schedules

which separate these groups e z ante when each risk type chooses the

schedule which maximizes its utility. But as Dionne remarks, it is not

obvious that these schedules will be sustainable if persons with unfavor-

able experience can switch from firm to firm without being compelled to

provide a valid history of their experience. That is, if a high risk's hstory

does not necessarily "follow" him from firm to firm, optimal equilibrium

may not be sustainable.

There needs to be further development of the theory for such

consumerforesight models. At the same time, there needs to be further

empirical verification of the degree of foresight inhviduals actually

display. Do purchasers of automobile insurance know and fully take

account of the way their premiums will vary with their claims history? Do

workers know and take into account the way their future wages will vary

with observed productivity? If consumers display only limited foresight,

the models of market equilibrium developed in this paper will be

appropriate.

Akerlof, G. 1970. "The market for lemons: quality uncertainty and the

market mechanisms", Quur te~ ly Journal of Economics, 84:488-500.

Beard, R., T. Pentikainen, and E. Pesonen. 1979. Risk Theory, 2nd edi-

tion. London: Chapman and Hall (section 5.8).

Dionne, G. 1981. "Adverse Selection and Repeated Insurance Contrasts,"

Cahier 8/39, Departement de Science Economique, Universite de

Montreal, July.

Kunreuther, H. 1976. "Limited Knowledge and Insurance Protection,"

Public Policy, 24: 227-261.

Miyasaki, H. 1977. "The rat race and internal labor markets." The Bell

Journal of Economics, 833941418.

Pauly, M. 1974. "Over Insurance and Public Provision of Insurance: The

Roles of Moral Hazard and Adverse Selection," Quarterly Journal of

Economics , 88344-62.

Radner, R. 1981. "Monitoring Cooperative Agreements in a Repeated

Principal-Agent Relationship," Economet r i ca , 49: 1127-49.

Rothschild, M., and J. Stiglitz. 1976. "Equilibrium in competitive

insurance markets: An essay in the economics of imperfect informa-

tion." Quar te r l y Journa l of Economics , 90:629-649.

Rubinstein, A., and M.E. Yaari. 1980. "Repeated Insurance Contracts and

Moral Hazard," Research Memorandum No. 37, Center for Research in

Mathematical Economics and Game Theory, the Hebrew University,

Jerusalem.

Spence, M. 1978. "Product differentiation and consumer choice in

insurance markets," J m r n d of Publ ic Economics, 10:427-447.

Wilson, C. 1977. "A model of insurance markets with incomplete informa-

tion," Journa l of Economic Theory , 16:167-207.

Risk averse consumers of each type i with wealth, 4, want to choose a

value of Qi given iP i and Pi which maximizes:

E [u i (~ i ) I = Qi U i [A , - X + ( 1 - P i ) Qi] + Ui (A, - Pi Qi) (A .1 )

subject to

Let Ri be the contingency price ratio

Pi ( 1 4,) R' = ( l - P i ) B i

and define Rim= and R~~~~ as the values of Ri where Qi = 0 and Qi = X

respectively when one maximizes E [ Ui ( Q i ) ] without any constraint on 4.

*A more detailed discussion of this model appears in Kunreuther (1076).

Then if

t d u i ui = - d2Ui > 0 and U: = - < 0

d Qi d Q: the optimal solution to (A. 1) is given by:

Whenever Pi c 9 i , then Qi = X, since in this range the premium is

either actuarially fair or subsidized. Suppose both consumer types have

identical utility functions given by the exponential

UH(Y) = UL ( Y ) = -B''', where c is the risk aversion coefficient. Then Qi

is determined by